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Asymptotic Coefficients and Errors for Chebyshev Polynomial Approximations with Weak Endpoint Singularities: Effects of Different Bases
When solving differential equations by a spectral method, it is often
convenient to shift from Chebyshev polynomials with coefficients
to modified basis functions that incorporate the boundary conditions.
For homogeneous Dirichlet boundary conditions, , popular choices
include the ``Chebyshev difference basis", with coefficients here denoted and the ``quadratic-factor
basis functions" with coefficients
. If is weakly singular at the boundaries, then will
decrease proportionally to for some positive
constant , where the is a logarithm or a constant. We prove that
the Chebyshev difference coefficients decrease more slowly by a factor
of while the quadratic-factor coefficients decrease more slowly
still as . The error for the unconstrained
Chebyshev series, truncated at degree , is
in the interior, but is worse by one power of
in narrow boundary layers near each of the endpoints. Despite having nearly
identical error \emph{norms}, the error in the Chebyshev basis is concentrated
in boundary layers near both endpoints, whereas the error in the
quadratic-factor and difference basis sets is nearly uniform oscillations over
the entire interval in . Meanwhile, for Chebyshev polynomials and the
quadratic-factor basis, the value of the derivatives at the endpoints is
, but only for the difference basis